A089983 1, 1, 1, 1, ... a, b, c, d, ab-cd, ...

1, 1, 1, 1, 0, 1, 1, -1, 1, 2, -3, 5, 17, -91, 1532, 139497, -213710951, 29812036392235, 6371158648631364574889, -189937213493701003981668660072118562, 1210120120447335073097142485947209203511752911347585124133

Offset: 1

Author

Ray Chandler, following a suggestion of Rainer Rosenthal, Nov 18 2003

Keywords

• sign
• easy

Comments

Inspired by the formula for the determinant of a 2 X 2 matrix.

Sequence b(n,p) = a(n) (mod p), p prime, is a periodic sequence. Letting l(p) denotes the length of the period of b(n,p) we get l(2)=5, l(3)=11, l(5)=31... Is there any rule for l(p) ? - Benoit Cloitre, Nov 19 2003

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..26

Crossrefs

Cf. A089984.

Programs

Magma

I:=[1,1,1,1]; [n le 4 select I[n] else -Self(n-1)*Self(n-2)+Self(n-3)*Self(n-4): n in [1..22]]; // Vincenzo Librandi, Mar 30 2014

Mathematica

nxt[{a_,b_,c_,d_}]:={b,c,d,a b-c d}; NestList[nxt,{1,1,1,1},20][[All,1]] (* Harvey P. Dale, Oct 30 2021 *)

PARI

a=b=c=d=1;for(n=5,20,e=a*b-c*d;a=b;b=c;c=d;d=e;print1(e,","))

Formula

a(1)=a(2)=a(3)=a(4)=1, for n>4 a(n)=a(n-4)*a(n-3)-a(n-2)*a(n-1).

a(n) is asymptotic (in absolute value) to A^(phi^n) where phi=golden ratio and A=1.005384.. (follows same kind of behavior as A000301, A007660) - Benoit Cloitre, Nov 19 2003